# Maxwell's equations

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In classical electromagnetism, Maxwell's equations are a set of four partial differential equations that describe the properties of the electric and magnetic fields and relate them to their sources, charge density and current density. James Clerk Maxwell used the equations to show that light is an electromagnetic wave. Individually, the equations are known as Gauss' law, Gauss' law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction.

These four equations, together with the Lorentz force law (derived by Maxwell),[1] are the complete set of laws of classical electromagnetism.

## General formulations of Maxwell's equations

The equations in this section are given in SI units. Unlike the equations of mechanics (for example), Maxwell's equations are not unchanged in other unit systems. Though the general form remains the same, various definitions get changed and different constants appear at different places. Other than SI (used in engineering), the units commonly used are Gaussian units (based on the cgs system and considered to have some theoretical advantages over SI[2]), Lorentz-Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics). See below for CGS-Gaussian units.

Two equivalent, general formulations of Maxwell's equations follow. The first separates bound charge and bound current (which arise in the context of dielectric and/or magnetized materials) from free charge and free current (the more conventional type of charge and current). This separation is useful for calculations involving dielectric or magnetized materials. The second formulation treats all charge equally, combining free and bound charge into total charge (and likewise with current). This is the more fundamental or microscopic point of view, and is particularly useful when no dielectric or magnetic material is present. More detail, and a proof that these two formulations are mathematically equivalent, are given in section 3.

Symbols in bold represent vector quantities, whereas symbols in italics represent scalar quantities. The definitions of terms used in the two tables of equations are given in another table immediately following.

##### Table 1: Formulation in terms of free charge and current
Name Differential form Integral form
Gauss's law: ${\displaystyle \nabla \cdot \mathbf {D} =\rho _{f}}$ ${\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =Q_{f,S}}$
Gauss's law for magnetism: ${\displaystyle \nabla \cdot \mathbf {B} =0}$ ${\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0}$
Maxwell-Faraday equation</br>(Faraday's law of induction): ${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$ ${\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {\partial \Phi _{B,S}}{\partial t}}}$
Ampère's Circuital Law
(with Maxwell's correction):
${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}}$ ${\displaystyle \oint _{\partial S}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =I_{f,S}+{\frac {\partial \Phi _{D,S}}{\partial t}}}$
##### Table 2: Formulation in terms of total charge and current
Name Differential form Integral form
Gauss's law: ${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}}$ ${\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {Q_{S}}{\epsilon _{0}}}}$
Gauss's law for magnetism: ${\displaystyle \nabla \cdot \mathbf {B} =0}$ ${\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0}$
Maxwell-Faraday equation</br>(Faraday's law of induction): ${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$ ${\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {\partial \Phi _{B,S}}{\partial t}}}$
Ampère's Circuital Law
(with Maxwell's correction):
${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\ }$ ${\displaystyle \ }$ ${\displaystyle \oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}I_{S}+\mu _{0}\epsilon _{0}{\frac {\partial \Phi _{E,S}}{\partial t}}}$

The following table provides the meaning of each symbol and the SI unit of measure:

##### Table 3: Definitions and units
Symbol Meaning (first term is the most common) SI Unit of Measure
${\displaystyle \mathbf {\nabla \cdot } }$ the divergence operator per meter (factor contributed by applying either operator)
${\displaystyle \mathbf {\nabla \times } }$ the curl operator
${\displaystyle {\frac {\partial }{\partial t}}}$ partial derivative with respect to time per second (factor contributed by applying the operator)
${\displaystyle \mathbf {E} \ }$ electric field volt per meter or, equivalently,
newton per coulomb
${\displaystyle \mathbf {B} \ }$ magnetic field
also called the magnetic induction
also called the magnetic field density
also called the magnetic flux density
tesla, or equivalently,
weber per square meter
voltsecond per square meter
${\displaystyle \mathbf {D} \ }$ electric displacement field coulombs per square meter or, equivalently,
newton per volt-meter
${\displaystyle \mathbf {H} \ }$ magnetizing field
also called auxiliary magnetic field
also called magnetic field intensity
also called magnetic field
ampere per meter
${\displaystyle \epsilon _{0}\ }$ permittivity of free space, officially the electric constant,</br> a universal constant farads per meter
${\displaystyle \mu _{0}\ }$ permeability of free space, officially the magnetic constant,</br> a universal constant henries per meter, or newtons per ampere squared
${\displaystyle \ \rho _{f}\ }$ free charge density (not including bound charge) coulomb per cubic meter
${\displaystyle \ \rho \ }$ total charge density (including both free and bound charge) coulomb per cubic meter
${\displaystyle \oint _{S}\mathbf {E\cdot \mathrm {d} A} }$ the flux of the electric field over any closed gaussian surface S joule-meter per coulomb
${\displaystyle Q_{f,S}\ }$ net unbalanced free electric charge enclosed by the</br>Gaussian surface S (not including bound charge) coulombs
${\displaystyle Q_{S}\ }$ net unbalanced electric charge enclosed by the Gaussian</br>surface S (including both free and bound charge) coulombs
${\displaystyle \oint _{S}\mathbf {B\cdot \mathrm {d} A} }$ the flux of the magnetic field over any closed surface S tesla meter-squared or weber
${\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} }$ line integral of the electric field along the boundary ∂S</br>(therefore necessarily a closed curve) of the surface S joule per coulomb
${\displaystyle \Phi _{B,S}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} }$ magnetic flux over any surface S (not necessarily closed) weber
${\displaystyle \mathbf {J} _{f}}$ free current density (not including bound current) ampere per square meter
${\displaystyle \mathbf {J} }$ total current density (including both free and bound current) ampere per square meter
${\displaystyle \oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} }$ line integral of the magnetic field over</br> the closed boundary ∂S of the surface S tesla-meter
${\displaystyle I_{f,S}=\int _{S}\mathbf {J} _{f}\cdot \mathrm {d} \mathbf {A} }$ net free electrical current passing through</br>the surface S (not including bound current) amperes
${\displaystyle I_{S}=\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {A} }$ net electrical current passing through the</br>surface S (including both free and bound current) amperes
${\displaystyle \Phi _{E,S}=\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} }$ electric flux through any surface S, not necessarily closed joule-meter per coulomb
${\displaystyle \Phi _{D,S}=\int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} }$ flux of electric displacement field through any surface S, not necessarily closed coulombs
${\displaystyle \mathrm {d} \mathbf {A} }$ differential vector element of surface area A, with infinitesimally

small magnitude and direction normal to surface S

square meters
${\displaystyle \mathrm {d} \mathbf {l} }$ differential vector element of path length tangential to contour meters

Maxwell's equations are generally applied to macroscopic averages of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms (where they undergo quantum mechanical effects as well). It is only in this averaged sense that one can define quantities such as the permittivity and permeability of a material. At the microscopic level, Maxwell's equations, ignoring quantum effects, describe fields, charges and currents in free space — but at this level of detail one must include all charges, even those at an atomic level, generally an intractable problem.

## History

Although James Clerk Maxwell was not the originator of these equations, he nevertheless derived three of them again independently in conjunction with his molecular vortex model of Faraday's "lines of force", along with the full version of Faraday's law of induction. In doing so he made an important addition to Ampère's circuital law.

Maxwell also developed Faraday's law of induction into another equation, which used to be listed as a 'Maxwell's equation' but is nowadays known as the Lorentz force law.[3]

### The term Maxwell's equations

Controversy has always surrounded the term Maxwell's equations concerning the extent to which Maxwell himself was involved in these equations. The term Maxwell's equations nowadays applies to a set of four equations that were grouped together as a distinct set in 1884 by Oliver Heaviside, in conjunction with Willard Gibbs.

The importance of Maxwell's role in these equations lies in the correction he made to Ampère's circuital law in his 1861 paper On Physical Lines of Force. He added the displacement current term to Ampère's circuital law and this enabled him to derive the electromagnetic wave equation in his later 1865 paper A Dynamical Theory of the Electromagnetic Field and demonstrate the fact that light is an electromagnetic wave. This fact was then later confirmed experimentally by Heinrich Hertz in 1887.

The reason that these equations are called Maxwell's equations is disputed. Some say that these equations were originally called the Heaviside-Hertz equations but that Einstein for whatever reason later referred to them as the Maxwell-Hertz equations. see pages 110-112 of Nahin's book[4][5]

These equations are based on the works of James Clerk Maxwell, and Heaviside made no secret of the fact that he was working from Maxwell's papers. Heaviside aimed to produce a symmetrical set of equations that were crucial as regards deriving the telegrapher's equations. The net result was a set of four equations, three of which had appeared in substance throughout Maxwell's previous papers, in particular Maxwell's 1861 paper On Physical Lines of Force and 1865 paper A Dynamical Theory of the Electromagnetic Field. The fourth was a partial time derivative version of Faraday's law of induction that doesn't include motionally induced EMF. [6]

Of Heaviside's equations, the most important in deriving the telegrapher's equations was the version of Ampère's circuital law that had been amended by Maxwell in this 1861 paper to include what is termed the displacement current.

### Maxwell's On Physical Lines of Force (1861)

Three of Heaviside's four equations appeared throughout Maxwell's 1861 paper On Physical Lines of Force:

(i) At equation (56) of Maxwell's 1861 paper we see ${\displaystyle \nabla \cdot \mathbf {B} =0}$.

(ii) At equation (112) we see Ampère's circuital law with Maxwell's correction. It is this correction called displacement current which is the most significant aspect of Maxwell's work in electromagnetism as it enabled him to later derive the electromagnetic wave equation in his 1865 paper A Dynamical Theory of the Electromagnetic Field, and hence show that light is an electromagnetic wave. It is therefore this aspect of Maxwell's work which gives Heaviside's equations their full significance. (Interestingly, Kirchhoff derived the telegrapher's equations in 1857 without using displacement current. But he did use Poisson's equation and the equation of continuity which are the mathematical ingredients of the displacement current. Nevertheless, Kirchhoff believed his equations to be applicable only inside an electric wire and so he is not credited with having discovered that light is an electromagnetic wave).

(iii) At equation (113) we see Gauss's law.

(iv) Heaviside's fourth equation introduced a restricted partial time derivative version of Faraday's law of induction. (A full version of Faraday's law of induction had appeared at equation (54) of Maxwell's 1861 paper). It is important however to note that Heaviside's partial time derivative notation, as opposed to the total time derivative notation used by Maxwell at equations (54), resulted in the loss of the v × B term that appeared in Maxwell's equation (77). Nowadays, the v × B term appears in the force law F = q ( E + v × B ) which sits adjacent to Maxwell's equations and bears the name Lorentz force. The Lorentz Force corresponds in effect to Maxwell's equation (77), but it appeared in this paper when Lorentz was still a young boy.

### Maxwell's A Dynamical Theory of the Electromagnetic Field (1865)

Confusion over the term "Maxwell's equations" is further increased because it is also sometimes used for a set of eight equations that appeared in Part III of Maxwell's 1865 paper A Dynamical Theory of the Electromagnetic Field, entitled "General Equations of the Electromagnetic Field" [1] (page 480 of the article and page 2 of the pdf link), a confusion compounded by the writing of six of those eight equations as three separate equations (one for each of the Cartesian axes), resulting in twenty equations in twenty unknowns. (As noted above, this terminology is not common: Modern references to the term "Maxwell's equations" usually refer to the Heaviside restatements.)

These original eight equations are nearly identical to the Heaviside versions in substance, but they have some superficial differences. In fact, only one of the Heaviside versions is completely unchanged from these original equations, and that is Gauss's law (Maxwell's equation G below). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation A below) with Ampère's circuital law (equation C below). This amalgamation, which Maxwell himself originally made at equation (112) in his 1861 paper "On Physical Lines of Force" (see above), is the one that modifies Ampère's circuital law to include Maxwell's displacement current.

The eight original Maxwell's equations can be written in modern vector notation as follows:

(A) The law of total currents
${\displaystyle \mathbf {J} _{tot}=\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}}$
(B) The equation of magnetic force
${\displaystyle \mu \mathbf {H} =\nabla \times \mathbf {A} }$
(C) Ampère's circuital law
${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{tot}}$
(D) Electromotive force created by convection, induction, and by static electricity. (This is in effect the Lorentz force)
${\displaystyle \mathbf {E} =\mu \mathbf {v} \times \mathbf {H} -{\frac {\partial \mathbf {A} }{\partial t}}-\nabla \phi }$
(E) The electric elasticity equation
${\displaystyle \mathbf {E} ={\frac {1}{\epsilon }}\mathbf {D} }$
(F) Ohm's law
${\displaystyle \mathbf {E} ={\frac {1}{\sigma }}\mathbf {J} }$
(G) Gauss's law
${\displaystyle \nabla \cdot \mathbf {D} =\rho }$
(H) Equation of continuity
${\displaystyle \nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}}$
Notation
${\displaystyle \mathbf {H} }$ is the magnetizing field, which Maxwell called the "magnetic intensity".
${\displaystyle \mathbf {J} }$ is the electric current density (with ${\displaystyle \mathbf {J} _{tot}}$ being the total current including displacement current).
${\displaystyle \mathbf {D} }$ is the displacement field (called the "electric displacement" by Maxwell).
${\displaystyle \rho }$ is the free charge density (called the "quantity of free electricity" by Maxwell).
${\displaystyle \mathbf {A} }$ is the magnetic vector potential (called the "angular impulse" by Maxwell).
${\displaystyle \mathbf {E} }$ is called the "electromotive force" by Maxwell. The term electromotive force is nowadays used for voltage, but it is clear from the context that Maxwell's meaning corresponded more to the modern term electric field.
${\displaystyle \Phi }$ is the electric potential (which Maxwell also called "electric potential").
${\displaystyle \sigma }$ is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).

It is interesting to note the ${\displaystyle \mu \mathbf {v} \times \mathbf {H} }$ term that appears in equation D. Equation D is therefore effectively the Lorentz force, similarly to equation (77) of his 1861 paper (see above).

When Maxwell derives the electromagnetic wave equation in his 1865 paper, he uses equation D to cater for electromagnetic induction rather than Faraday's law of induction which is used in modern textbooks. (Faraday's law itself does not appear among his equations.) However, Maxwell drops the ${\displaystyle \mu \mathbf {v} \times \mathbf {H} }$ term from equation D when he is deriving the electromagnetic wave equation, as he considers the situation only from the rest frame.

## Details and special cases of Maxwell's equations

### Bound charge and current

If an electric field is applied to a dielectric material, each of the molecules responds by forming a microscopic dipole -- its atomic nucleus will move a tiny distance in the direction of the field, while its electrons will move a tiny distance in the opposite direction. This is called polarization of the material. The distribution of charge that results from these tiny movements turn out to be identical to having a layer of positive charge on one side of the material, and a layer of negative charge on the other side -- a macroscopic separation of charge, even though all of the charges involved are "bound" to a single molecule. This is called bound charge. Likewise, in a magnetized material, there is effectively a "bound current" circulating around the material, despite the fact that no individual charge is travelling a distance larger than a single molecule.

#### Proof that the two general formulations are equivalent

In this section, a simple proof is outlined which shows that the two alternate general formulations of Maxwell's equations given in Section 1 are mathematically equivalent.

The relation between polarization, magnetization, bound charge, and bound current is as follows:

${\displaystyle \rho _{b}=-\nabla \cdot \mathbf {P} }$
${\displaystyle \mathbf {J} _{b}=\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}}$
${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} +\mathbf {P} }$
${\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )}$
${\displaystyle \rho =\rho _{b}+\rho _{f}\ }$
${\displaystyle \mathbf {J} =\mathbf {J} _{b}+\mathbf {J} _{f}}$

where P and M are polarization and magnetization, and ρb and Jb are bound charge and current, respectively. Plugging in these relations, it can be easily demonstrated that the two formulations of Maxwell's equations given in Section 1 are precisely equivalent.

### Constitutive relations

In order to apply Maxwell's equations (the formulation in terms of free charge and current, and D and H), it is necessary to specify the relations between D and E, and H and B. These are called constitutive relations, and correspond physically to specifying the response of bound charge and current to the field, or equivalently, how much polarization and magnetization a material acquires in the presence of electromagnetic fields.

#### Case without magnetic or dielectric materials

In the absence of magnetic or dielectric materials, the relations are simple:

${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} ,\;\;\;\mathbf {H} =\mathbf {B} /\mu _{0}}$

where ε0 and μ0 are two universal constants, called the permittivity of free space and permeability of free space, respectively.

#### Case of linear materials

In a "linear", isotropic, nondispersive, uniform material, the relations are also straightforward:

${\displaystyle \mathbf {D} =\epsilon \mathbf {E} ,\;\;\;\mathbf {H} =\mathbf {B} /\mu }$

where ε and μ are constants (which depend on the material), called the permittivity and permeability, respectively, of the material.

#### General case

For real-world materials, the constitutive relations are not simple proportionalities, except approximately. The relations can usually still be written:

${\displaystyle \mathbf {D} =\epsilon \mathbf {E} ,\;\;\;\mathbf {H} =\mathbf {B} /\mu }$

but ε and μ are not, in general, simple constants, but rather functions. For example, ε and μ can depend upon:

• The strength of the fields (the case of nonlinearity, which occurs when ε and μ are functions of E and B; see, for example, Kerr and Pockels effects),
• The direction of the fields (the case of anisotropy, birefringence, or dichroism; which occurs when ε and μ are second-rank tensors),
• The frequency with which the fields vary (the case of dispersion, which occurs when ε and μ are functions of frequency; see, for example, Kramers–Kronig relations),
• The position inside the material (the case of a nonuniform material, which occurs when ε and μ vary from point to point within the material; for example in a domained structure, heterostructure or a liquid crystal),
• The history of the fields (the case of hysteresis, which occurs when ε and μ are functions of both present and past values of the fields).

#### Maxwell's equations in terms of E and B for linear materials

Substituting in the constitutive relations above, Maxwell's equations in a linear material (differential form only) are:

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{f}}{\epsilon }}}$
${\displaystyle \nabla \cdot \mathbf {B} =0}$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {B} =\mu \mathbf {J} _{f}+\mu \epsilon {\frac {\partial \mathbf {E} }{\partial t}}.}$

These are formally identical to the general formulation in terms of E and B (given above), except that the permittivity of free space was replaced with the permittivity of the material (see also displacement field, electric susceptibility and polarization density), the permeability of free space was replaced with the permeability of the material (see also magnetization, magnetic susceptibility and magnetic field), and only free charges and currents are included (instead of all charges and currents).

### Maxwell's equations in vacuum

See also: Electromagnetic wave equation and Sinusoidal plane-wave solutions of the electromagnetic wave equation

Starting with the equations appropriate in the case without dielectric or magnetic materials, and assuming that there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space:

${\displaystyle \nabla \cdot \mathbf {E} =0}$
${\displaystyle \nabla \cdot \mathbf {B} =0}$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {B} =\ \ \mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$

These equations have a solution in terms of traveling sinusoidal plane waves, with the electric and magnetic field directions orthogonal to one another and the direction of travel, and with the two fields in phase, traveling at the speed[7]

${\displaystyle c_{0}={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}\ .}$

In fact, Maxwell's equations explains specifically how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law. That electric field, in turn, creates a changing magnetic field through Maxwell's correction to Ampère's law. This perpetual cycle allows these waves, known as electromagnetic radiation, to move through space, always at velocity c0.

Maxwell discovered that this quantity c0 equals the speed of light in vacuum (known from early experiments), and concluded (correctly) that light is a form of electromagnetic radiation.

### With magnetic monopoles

Maxwell's equations of electromagnetism relate the electric and magnetic fields to the motions of electric charges. The standard form of the equations provide for an electric charge, but posit no magnetic charge. Except for this, the equations are symmetric under interchange of electric and magnetic field. In fact, symmetric equations can be written when all charges are zero, and this is how the wave equation is derived (see immediately above).

Fully symmetric equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.[2] With the inclusion of a variable for these magnetic charges, say ${\displaystyle \rho _{m}\,}$, there will also be "magnetic current" variable in the equations, ${\displaystyle {\vec {J}}_{m}\,}$. The extended Maxwell's equations, simplified by nondimensionalization, are as follows:

Name Without Magnetic Monopoles With Magnetic Monopoles (hypothetical)
Gauss's law: ${\displaystyle {\vec {\nabla }}\cdot {\vec {E}}=4\pi \rho _{e}}$ ${\displaystyle {\vec {\nabla }}\cdot {\vec {E}}=4\pi \rho _{e}}$
Gauss' law for magnetism: ${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$ ${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=4\pi \rho _{m}}$
${\displaystyle -{\vec {\nabla }}\times {\vec {E}}={\frac {\partial {\vec {B}}}{\partial t}}}$ ${\displaystyle -{\vec {\nabla }}\times {\vec {E}}={\frac {\partial {\vec {B}}}{\partial t}}+4\pi {\vec {j}}_{m}}$
Ampère's law
(with Maxwell's extension):
${\displaystyle {\vec {\nabla }}\times {\vec {B}}={\frac {\partial {\vec {E}}}{\partial t}}+4\pi {\vec {j}}_{e}}$  ${\displaystyle \ }$    ${\displaystyle {\vec {\nabla }}\times {\vec {B}}={\frac {\partial {\vec {E}}}{\partial t}}+4\pi {\vec {j}}_{e}}$
Note: the Bivector notation embodies the sign swap, and these four equations can be written as only one equation.

If magnetic charges do not exist, or if they exist but where they are not present in a region, then the new variables are zero, and the symmetric equations reduce to the conventional equations of electromagnetism such as ${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$. Classically, the question is "Why does the magnetic charge always seem to be zero?"

## Materials and dynamics

The fields in Maxwell's equations are generated by charges and currents. Conversely, the charges and currents are affected by the fields through the Lorentz force equation:

${\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ),}$

where q is the charge on the particle and v is the particle velocity. (It also should be remembered that the Lorentz force is not the only force exerted upon charged bodies, which also may be subject to gravitational, nuclear, etc. forces.) Therefore, in both classical and quantum physics, the precise dynamics of a system form a set of coupled differential equations, which are almost always too complicated to be solved exactly, even at the level of statistical mechanics.[8] This remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents, which enter Maxwell's equations through the constitutive equations, as described next.

Commonly, real materials are approximated as "continuum" media with bulk properties such as the refractive index, permittivity, permeability, conductivity, and/or various susceptibilities. These lead to the macroscopic Maxwell's equations, which are written (as given above) in terms of free charge/current densities and D, H, E, and B ( rather than E and B alone ) along with the constitutive equations relating these fields. For example, although a real material consists of atoms whose electronic charge densities can be individually polarized by an applied field, for most purposes behavior at the atomic scale is not relevant and the material is approximated by an overall polarization density related to the applied field by an electric susceptibility.

Continuum approximations of atomic-scale inhomogeneities cannot be determined from Maxwell's equations alone. but require some type of quantum mechanical analysis such as quantum field theory as applied to condensed matter physics. See, for example, density functional theory, Green–Kubo relations and Green's function (many-body theory). Various approximate transport equations have evolved, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier-Stokes equations. Some examples where these equations are applied are magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, plasma modeling. An entire physical apparatus for dealing with these matters has developed. A different set of homogenization methods (evolving from a tradition in treating materials such as conglomerates and laminates) are based upon approximation of an inhomogeneous material by a homogeneous effective medium[9][10] (valid for excitations with wavelengths much larger than the scale of the inhomogeneity).[11][12][13][14]

Theoretical results have their place, but often require fitting to experiment. Continuum-approximation properties of many real materials rely upon measurement,[15] for example, ellipsometry measurements.

In practice, some materials properties have a negligible impact in particular circumstances, permitting neglect of small effects. For example: optical nonlinearities can be neglected for low field strengths; material dispersion is unimportant where frequency is limited to a narrow bandwidth; material absorption can be neglected for wavelengths where a material is transparent; and metals with finite conductivity often are approximated at microwave or longer wavelengths as perfect metals with infinite conductivity (forming hard barriers with zero skin depth of field penetration).

And, of course, some situations demand that Maxwell's equations and the Lorentz force be combined with other forces that are not electromagnetic. An obvious example is gravity. A more subtle example, which applies where electrical forces are weakened due to charge balance in a solid or a molecule, is the Casimir force from quantum electrodynamics.[16]

The connection of Maxwell's equations to the rest of the physical world is via the fundamental sources of charges and currents and the forces on them, and also by the properties of physical materials.

### Boundary conditions

Although Maxwell's equations apply throughout space and time, practical problems are finite and solutions to Maxwell's equations inside the solution region are joined to the remainder of the universe through boundary conditions[17][18] [19] and started in time using initial conditions.[20] In some cases, like waveguides or cavity resonators, the solution region is largely isolated from the universe, for example, by metallic walls, and boundary conditions at the walls define the fields with influence of the outside world confined to the input/output ends of the structure.[21] In other cases, the universe at large sometimes is approximated by an artificial absorbing boundary,[22][23][24] or, for example for radiating antennas or communication satellites, these boundary conditions can take the form of asymptotic limits imposed upon the solution.[25] In addition, for example in an optical fiber or thin-film optics, the solution region often is broken up into subregions with their own simplified properties, and the solutions in each subregion must be joined to each other across the subregion interfaces using boundary conditions.[26][27][28] Following are some links of a general nature concerning boundary value problems: Examples of boundary value problems, Sturm-Liouville theory, Dirichlet boundary condition, Neumann boundary condition, mixed boundary condition, Cauchy boundary condition, Sommerfeld radiation condition. Needless to say, one must choose the boundary conditions appropriate to the problem being solved. See also Kempel[29] and the extraordinary book by Friedman.[30]

## The Heaviside versions in detail

### Gauss's law

Gauss's law describes the relation between the electric field and the distribution of electric charge, as follows:

${\displaystyle \nabla \cdot \mathbf {D} =\rho _{f}\ .}$

The formulation of Table 1 is assumed; that is, ρf is the "free" electric charge density (in units of C/m³), not including bound charge from the polarization of a material, and ${\displaystyle \mathbf {D} }$ is the electric displacement field (in units of C/m²). For stationary charges in vacuum, the force exerted upon one point charge by another as found from Gauss's law is Coulomb's law.

The equivalent integral form (by the divergence theorem) of Gauss' law is:

${\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =Q_{\mathrm {enclosed} }\ }$

where:

S is any fixed, closed surface,
The integral is a surface integral, i.e. ${\displaystyle \mathrm {d} \mathbf {A} }$ is a vector whose magnitude is the area of a differential square on the closed surface A, and whose direction is an outward-facing normal vector, and
${\displaystyle Q_{\mathrm {enclosed} }}$ is the free charge enclosed within the surface S. (If the surface itself is charged, that gives an extra contribution weighted by a factor 1/2.)

In a linear, isotropic, homogeneous material, with instantaneous response to field changes, ${\displaystyle \mathbf {D} }$ is directly related to the electric field ${\displaystyle \mathbf {E} }$ via a material-dependent constant called the permittivity, ${\displaystyle \epsilon }$:

${\displaystyle \mathbf {D} =\varepsilon \mathbf {E} }$.

The material permittivity ${\displaystyle \epsilon }$ can also be written as ε0 εr where εr is the material's relative permittivity or its dielectric constant. No material (except free space) is precisely linear and isotropic, but many materials are approximately so. The permittivity of free space, or electric constant, is denoted as ${\displaystyle \epsilon _{0}}$ (approximately 8.854 pF/m), and appears in:

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{t}}{\varepsilon _{0}}}}$

where, again, ${\displaystyle \mathbf {E} }$ is the electric field (in units of V/m), ${\displaystyle \rho _{t}}$ is the total charge density (including bound charges). The formulation of Table 2 is assumed.

Some insight into Gauss' law is found using the Maxwell-Faraday equation:

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\ ,}$

which shows the solenoidal portion of E is determined by the time variation of the magnetic field. Thus, in electrostatics (that is, when the system is unchanging in time), by Helmholtz decomposition the E-field can be expressed in terms of a scalar field as:

${\displaystyle \mathbf {E} (\mathbf {r} )=-\nabla \phi (\mathbf {r} )\ .}$

Time independence not only allows E to be expressed as a gradient, but also removes any time-delay in material response (ε independent of time), so the equation determining the electrostatic potential ɸ (r ) is:

${\displaystyle \nabla \cdot \mathbf {D} (\mathbf {r} )=-\nabla \cdot \left(\epsilon (\mathbf {r} )\nabla \phi (\mathbf {r} )\right)=\rho _{f}(\mathbf {r} )\ ,}$

which is Poisson's equation in the case where ε is independent of position (that is, when the material is homogeneous). The formulation of Table 1 is assumed. That is, the bound charge is subsumed under the permittivity, and only the free charge is explicit on the right side of the equation.

### Gauss's law for magnetism

"Gauss's law for magnetism" states that the divergence of the magnetic field is always zero (in other words, the magnetic field is a solenoidal vector field):

${\displaystyle \nabla \cdot \mathbf {B} =0\ ,}$

where ${\displaystyle \mathbf {B} }$ is the magnetic B-field (in units of tesla, denoted "T"), also called "magnetic flux density", "magnetic induction", or simply "magnetic field". It is interpreted as saying there is no "magnetic" charge that is the analog of the electric charge, and often this equation is referred to as "the absence of magnetic monopoles". Differently put, the basic entity for magnetism is the magnetic dipole, which orients itself in a magnetic field.

By the divergence theorem, the above divergence equation has an equivalent integral form:

${\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0\ ,}$

where ${\displaystyle \mathrm {d} \mathbf {A} }$ is an infinitesimal vector corresponding to the area of a differential square on the surface S with an outward facing surface normal defining its direction.

Like the electric field's integral form, this equation works only if the integral is done over a closed surface.

This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Structurally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. Hence, the above reference to this law as saying there are no magnetic monopoles.

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\ .}$

This equation is usually referred to as "Faraday's law of induction", but in fact it is only a restricted form of Faraday's law; for example, it doesn't apply to situations involving motionally induced EMF.

### The Maxwell-Ampère equation

Ampère's circuital law describes the source of the magnetic field,

${\displaystyle \nabla \times \mathbf {H} =\mathbf {j} +{\frac {\partial \mathbf {D} }{\partial t}}}$

where ${\displaystyle \mathbf {H} }$ is the magnetic field strength (in units of A/m), related to the magnetic flux density ${\displaystyle \mathbf {B} }$ by a constant called the permeability, μ (${\displaystyle \mathbf {B} =\mu \mathbf {H} }$), and ${\displaystyle \mathbf {j} }$ is the current density, defined by: ${\displaystyle \mathbf {j} =\rho _{q}\mathbf {v} }$ where ${\displaystyle \mathbf {v} }$ is a vector field called the drift velocity that describes the velocities of the charge carriers which have a density described by the scalar function ρq. The second term on the right hand side of Ampère's Circuital Law is known as the displacement current.

It was Maxwell who added the displacement current term to Ampère's Circuital Law at equation (112) in his 1861 paper On Physical Lines of Force.

Maxwell used the displacement current in conjunction with the original eight equations in his 1865 paper A Dynamical Theory of the Electromagnetic Field to derive a wave equation that has the velocity of light. Most modern textbooks derive this electromagnetic wave equation using the 'Heaviside Four'.

In free space, the permeability μ is the magnetic constant, μ0, which is defined to be exactly 4π×10-7 Wb/Am. Also, the permittivity becomes the electric constant ε0, also a defined quantity. Thus, in free space, the equation becomes:

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$

Using Stokes theorem the equivalent integral form can be found:

${\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} \mathbf {l} }$${\displaystyle =\mu _{0}\int _{S}\mathbf {j} \cdot \mathrm {d} \mathbf {A} +\mu _{0}\varepsilon _{0}\int _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {A} }$${\displaystyle =\mu _{0}I_{\mathrm {encircled} }+\mu _{0}\varepsilon _{0}\int _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {A} }$

C is the edge of the open surface A (any surface with the curve C as its edge will do), and Iencircled is the current encircled by the curve C (the current through any surface is defined by the equation: ${\displaystyle {\begin{matrix}I_{\mathrm {through} \ A}=\int _{S}\mathbf {j} \cdot \mathrm {d} \mathbf {A} \end{matrix}}}$). Sometimes this integral form of Ampere-Maxwell Law is written as:

${\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}(I_{\mathrm {enc} }+I_{\mathrm {d,enc} })}$     because the term     ${\displaystyle \varepsilon _{0}\int _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {A} }$

is displacement current. The displacement current concept was Maxwell's greatest innovation in electromagnetic theory. It implies that a magnetic field appears during the charge or discharge of a capacitor. If the electric flux density does not vary rapidly, the second term on the right hand side (the displacement flux) is negligible, and the equation reduces to Ampere's law.

## Maxwell's equations in CGS units

The above equations are given in the International System of Units, or SI for short. In a related unit system, called cgs (short for centimeter-gram-second), the equations take the following form:

${\displaystyle \nabla \cdot \mathbf {D} =4\pi \rho _{f}}$
${\displaystyle \nabla \cdot \mathbf {B} =0}$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {H} ={\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {J} _{f}}$

Where c is the speed of light in a vacuum. For the electromagnetic field in a vacuum, the equations become:

${\displaystyle \nabla \cdot \mathbf {E} =0}$
${\displaystyle \nabla \cdot \mathbf {B} =0}$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$
${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}}$

In this system of units the relation between magnetic induction, magnetic field and total magnetization take the form:

${\displaystyle \mathbf {B} =\mathbf {H} +4\pi \mathbf {M} }$

With the linear approximation:

${\displaystyle \mathbf {B} =(\ 1+4\pi \chi _{m}\ )\mathbf {H} }$

${\displaystyle \chi _{m}}$ for vacuum is zero and therefore:

${\displaystyle \mathbf {B} =\mathbf {H} }$

and in the ferro or ferri magnetic materials where ${\displaystyle \chi _{m}}$ is much bigger than 1:

${\displaystyle \mathbf {B} =4\pi \chi _{m}\mathbf {H} }$

The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:

${\displaystyle \mathbf {F} =q\left(\mathbf {E} +{\frac {\mathbf {v} }{c}}\times \mathbf {B} \right),}$

where ${\displaystyle q\ }$ is the charge on the particle and ${\displaystyle \mathbf {v} \ }$ is the particle velocity. This is slightly different from the SI-unit expression above. For example, here the magnetic field ${\displaystyle \mathbf {B} \ }$ has the same units as the electric field ${\displaystyle \mathbf {E} \ }$.

## Maxwell's equations and special relativity

Maxwell's equations have a close relation to special relativity: Not only were Maxwell's equations a crucial part of the historical development of special relativity, but also, special relativity has motivated a compact mathematical formulation Maxwell's equations, in terms of covariant tensors.

### Historical developments

Maxwell's electromagnetic wave equation only applied in what he believed to be the rest frame of the luminiferous medium because he didn't use the vXB term of his equation (D) when he derived it. Maxwell's idea of the luminiferous medium was that it comprised of aethereal vortices aligned solenoidally along their rotation axes.

The American scientist A.A. Michelson set out to determine the velocity of the earth through the luminiferous medium aether using a light wave interferometer that he had invented. When the Michelson-Morley experiment was conducted by Edward Morley and Albert Abraham Michelson in 1887, it produced a null result for the change of the velocity of light due to the Earth's motion through the hypothesized aether. This null result was in line with the theory that was proposed in 1845 by George Stokes which suggested that the aether was entrained with the Earth's orbital motion.

Hendrik Lorentz objected to Stokes' aether drag model and in along with George FitzGerald and Joseph Larmor, he suggested another approach. Both Larmor (1897) and Lorentz (1899, 1904) derived the Lorentz transformation (so named by Henri Poincaré) as one under which Maxwell's equations were invariant. Poincaré (1900) analyzed the coordination of moving clocks by exchanging light signals. He also established mathematically the group property of the Lorentz transformation (Poincaré 1905).

This culminated in Albert Einstein's revolutionary theory of special relativity, which postulated the absence of any absolute rest frame, dismissed the aether as unnecessary (a bold idea, which did not come to Lorentz nor to Poincaré), and established the invariance of Maxwell's equations in all inertial frames of reference, in contrast to the famous Newtonian equations for classical mechanics. But the transformations between two different inertial frames had to correspond to Lorentz' equations and not - as former believed - to those of Galileo (called Galilean transformations).[32] Indeed, Maxwell's equations played a key role in Einstein's famous paper on special relativity; for example, in the opening paragraph of the paper, he motivated his theory by noting that a description of a conductor moving with respect to a magnet must generate a consistent set of fields irrespective of whether the force is calculated in the rest frame of the magnet or that of the conductor.[3]

General relativity has also had a close relationship with Maxwell's equations. For example, Kaluza and Klein showed in the 1920s that Maxwell's equations can be derived by extending general relativity into five dimensions. This strategy of using higher dimensions to unify different forces continues to be an active area of research in particle physics.

### Covariant formulation of Maxwell's equations

In special relativity, in order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form (cgs units).

One ingredient in this formulation is the electromagnetic tensor, a rank-2 antisymmetric tensor combining the electric and magnetic fields:

${\displaystyle F=\left({\begin{matrix}0&{\frac {-E_{x}}{c}}&{\frac {-E_{y}}{c}}&{\frac {-E_{z}}{c}}\\{\frac {E_{x}}{c}}&0&-B_{z}&B_{y}\\{\frac {E_{y}}{c}}&B_{z}&0&-B_{x}\\{\frac {E_{z}}{c}}&-B_{y}&B_{x}&0\end{matrix}}\right).}$

(Here SI units are used; in cgs units, one would have to replace c by 1.)

The other ingredient is the four-current: ${\displaystyle J^{\alpha }=(c\rho ,{\vec {J}})}$ where ${\displaystyle \rho }$ is the charge density and J is the current density.

With these ingredients, Maxwell's equations can be written:

${\displaystyle {4\pi \over c}j^{\beta }={\partial F^{\alpha \beta } \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }F^{\alpha \beta }\ {\stackrel {\mathrm {def} }{=}}\ {F^{\alpha \beta }}_{,\alpha }\,\!}$,

and

${\displaystyle 0=\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }\ {\stackrel {\mathrm {def} }{=}}\ {F_{\alpha \beta }}_{,\gamma }+{F_{\gamma \alpha }}_{,\beta }+{F_{\beta \gamma }}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \epsilon _{\delta \alpha \beta \gamma }{F^{\beta \gamma }}_{,\alpha }}$

where ${\displaystyle \,\epsilon _{\alpha \beta \gamma \delta }}$ is the Levi-Civita symbol, and

${\displaystyle {\partial \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }\ {\stackrel {\mathrm {def} }{=}}\ {}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {\partial }{\partial ct}},\nabla \right)}$

is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations. Upper and lower components of a vector, ${\displaystyle v^{\alpha }}$ and ${\displaystyle v_{\alpha }}$ respectively, are interchanged with the fundamental matrix g, e.g., g=diag(+1,-1,-1,-1).

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and Ampere's law with Maxwell's correction. The second equation is an expression of the two homogeneous equations, Faraday's law of induction and Gauss's law for magnetism.

Alternative covariant presentations of Maxwell's equations also exist, for example in terms of the four-potential; see Covariant formulation of classical electromagnetism for details.

## Maxwell's equations in terms of potentials

Maxwell's equations can be written in an alternative form, involving the electric potential (also called scalar potential) and magnetic potential (also called vector potential), as follows.[33] (The following equations are for vacuum only.)

First, Gauss' law for magnetism states:

${\displaystyle \nabla \cdot \mathbf {B} =0}$

by Helmholtz's theorem, B can be written in terms of a vector field A, called the magnetic potential:

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$

Second, plugging this into Faraday's law, we get:

${\displaystyle \nabla \times \left(\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}\right)=0}$

By Helmholtz's theorem, the quantity in parentheses can be written in terms of a scalar function ${\displaystyle \Phi }$, called the electric potential:

${\displaystyle \mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}=-\nabla \Phi }$

Combining these with the remaining two Maxwell's equations yields the four relations:

${\displaystyle \mathbf {E} =-\mathbf {\nabla } \Phi -{\frac {\partial \mathbf {A} }{\partial t}}}$
${\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} }$
${\displaystyle \nabla ^{2}\Phi +{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon _{0}}}}$
${\displaystyle \left(\nabla ^{2}\mathbf {A} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right)-\mathbf {\nabla } \left(\mathbf {\nabla } \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \Phi }{\partial t}}\right)=-\mu _{0}\mathbf {J} }$

These equations, taken together, are as powerful and complete as Maxwell's equations. Moreover, the problem has been reduced somewhat, as the electric and magnetic fields each have three components which need to be solved for (six components altogether), while the electric and magnetic potentials have only four components altogether. On the other hand, these equations appear more complicated than Maxwell's equations using just the electric and magnetic fields.

In fact, these equations can be simplified a good deal by taking advantage of gauge freedom—i.e., the fact that there are many different choices of A and ${\displaystyle \Phi }$ consistent with a given E and B. For more information, see the article gauge freedom.

## Maxwell's equations in terms of differential forms

In free space, where ε = ε0 and μ = μ0 are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. In what follows, cgs units, not SI units are used, however. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. Maxwell's equations then reduce to the Bianchi identity

${\displaystyle \mathrm {d} {\mathbf {F}}=0}$

where d denotes the exterior derivative — a natural coordinate and metric independent differential operator acting on forms — and the source equation

${\displaystyle \mathrm {d} *{\mathbf {F}}={\mathbf {J}}}$

where the (dual) Hodge star operator * is a linear transformation from the space of 2-forms to the space of (4-2)-forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric), and the fields are in natural units where ${\displaystyle 1/4\pi \epsilon _{0}=1}$. Here, the 3-form J is called the "electric current form" or "current 3-form" satisfying the continuity equation

${\displaystyle \mathrm {d} {\mathbf {J}}=0.}$

The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time if that region is a spacelike surface cross a timelike interval. As the exterior derivative is defined on any manifold, the differential form version of the Bianchi identity makes sense for any 4-dimensional manifold, whereas the source equation is defined if the manifold is oriented and has a Lorentz metric. In particular the differential form version of the Maxwell equations are a convenient and intuitive formulation of the Maxwell equations in general relativity.

In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call

${\displaystyle C:\Lambda ^{2}\ni {\mathbf {F}}\mapsto {\mathbf {G}}\in \Lambda ^{(4-2)}}$

the constitutive transformation. The role of this transformation is comparable to the Hodge duality transformation. The Maxwell equations in the presence of matter then become:

${\displaystyle \mathrm {d} {\mathbf {F}}=0}$
${\displaystyle \mathrm {d} {\mathbf {G}}={\mathbf {J}}}$

where the current 3-form J still satisfies the continuity equation dJ= 0.

When the fields are expressed as linear combinations (of exterior products) of basis forms ${\displaystyle {\mathbf {\theta }}^{p}}$,

${\displaystyle {\mathbf {F}}={\frac {1}{2}}F_{pq}{\mathbf {\theta }}^{p}\wedge {\mathbf {\theta }}^{q}}$.

the constitutive relation takes the form

${\displaystyle G_{pq}=C_{pq}^{mn}F_{mn}}$

where the field coefficient functions are antisymmetric in the indices and the constitutive coefficients are antisymmetric in the corresponding pairs. In particular, the Hodge duality transformation leading to the vacuum equations discussed above are obtained by taking

${\displaystyle C_{pq}^{mn}=g^{ma}g^{nb}\epsilon _{abpq}{\sqrt {-g}}}$

which up to scaling is the only invariant tensor of this type that can be defined with the metric.

In this formulation, electromagnetism generalises immediately to any 4-dimensional oriented manifold or with small adaptations any manifold, requiring not even a metric. Thus the expression of Maxwell's equations in terms of differential forms leads to a further notational and conceptual simplification. Whereas Maxwell's Equations could be written as two tensor equations instead of eight scalar equations, from which the propagation of electromagnetic disturbances and the continuity equation could be derived with a little effort, using differential forms leads to an even simpler derivation of these results.

#### Conceptual insight from this formulation

On the conceptual side, from the point of view of physics, this shows that the second and third Maxwell equations should be grouped together, be called the homogeneous ones, and be seen as geometric identities expressing nothing else than: the field F derives from a more "fundamental" potential A. While the first and last one should be seen as the dynamical equations of motion, obtained via the Lagrangian principle of least action, from the "interaction term" A J (introduced through gauge covariant derivatives), coupling the field to matter.

Often, the time derivative in the third law motivates calling this equation "dynamical", which is somewhat misleading; in the sense of the preceding analysis, this is rather an artifact of breaking relativistic covariance by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a kinetic term F *F for A; and take into account the non-physical degrees of freedom which can be removed by gauge transformation AA' = A-dα: see also gauge fixing and Fadeev-Popov ghosts.

## Classical electrodynamics as the curvature of a line bundle

An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or principal bundles with fibre U(1). The connection ${\displaystyle \nabla }$ on the line bundle has a curvature ${\displaystyle {\mathbf {F}}=\nabla ^{2}}$ which is a two-form that automatically satisfies ${\displaystyle \mathrm {d} {\mathbf {F}}=0}$ and can be interpreted as a field-strength. If the line bundle is trivial with flat reference connection d we can write ${\displaystyle \nabla =\mathrm {d} +{\mathbf {A}}}$ and F = dA with A the 1-form composed of the electric potential and the magnetic vector potential.

In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov-Bohm effect. In this experiment, a static magnetic field runs through a long magnetic wire (e.g. an Fe wire magnetized longitudinally). Outside of this wire the magnetic induction is zero, in contrast to the vector potential, which essentially depends on the magnetic flux through the cross-section of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor F = 0 throughout the space-time region outside the tube, during the experiment. This means by definition that the connection ${\displaystyle \nabla }$ is flat there.

However, as mentioned, the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantum-mechanically with a double-slit electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern. (See Michael Murray, Line Bundles, 2002 (PDF web link) for a simple mathematical review of this formulation. See also R. Bott, On some recent interactions between mathematics and physics, Canadian Mathematical Bulletin, 28 (1985) no. 2 pp 129-164.)

## Maxwell's equations in curved spacetime

Matter and energy generate curvature of spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum will also generate curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs units):

${\displaystyle {4\pi \over c}j^{\beta }=\partial _{\alpha }F^{\alpha \beta }+{\Gamma ^{\alpha }}_{\mu \alpha }F^{\mu \beta }+{\Gamma ^{\beta }}_{\mu \alpha }F^{\alpha \mu }\ {\stackrel {\mathrm {def} }{=}}\ D_{\alpha }F^{\alpha \beta }\ {\stackrel {\mathrm {def} }{=}}\ {F^{\alpha \beta }}_{;\alpha }\,\!}$,

and

${\displaystyle 0=\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }=D_{\gamma }F_{\alpha \beta }+D_{\beta }F_{\gamma \alpha }+D_{\alpha }F_{\beta \gamma }}$.

Here,

${\displaystyle {\Gamma ^{\alpha }}_{\mu \beta }\!}$

is a Christoffel symbol that characterizes the curvature of spacetime and ${\displaystyle D_{\gamma }}$ is the covariant derivative.

### Formulation in terms of differential forms

The formulation of the Maxwell equations in terms of differential forms can be used without change in general relativity. The equivalence of the more traditional general relativistic formulation using the covariant derivative with the differential form formulation can be seen as follows. Choose local coordinates ${\displaystyle x^{\alpha }}$ which gives a basis of 1-forms ${\displaystyle dx^{\alpha }}$ in every point of the open set where the coordinates are defined. Using this basis and cgs units we define

• The antisymmetric infinitesimal field tensor ${\displaystyle F_{\alpha \beta }}$, corresponding to the field 2-form F
${\displaystyle {\mathbf {F}}:={\frac {1}{2}}F_{\alpha \beta }\,\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }}$
• The current-vector infinitesimal 3-form J
${\displaystyle {\mathbf {J}}:={4\pi \over c}j^{\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }}$

Here g is as usual the determinant of the metric tensor ${\displaystyle g_{\alpha \beta }}$. A small computation that uses the symmetry of the Christoffel symbols (i.e. the torsion-freeness of the Levi Civita connection) and the covariant constantness of the Hodge star operator then shows that in this coordinate neighborhood we have:

• the Bianchi identity
${\displaystyle \mathrm {d} {\mathbf {F}}=2(\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma })\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }=0}$
• the source equation
${\displaystyle \mathrm {d} *{\mathbf {F}}={F^{\alpha \beta }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\beta \gamma \delta \eta }\mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }\wedge \mathrm {d} \,x^{\eta }={\mathbf {J}}}$
• the continuity equation
${\displaystyle \mathrm {d} {\mathbf {J}}={4\pi \over c}{j^{\alpha }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }=0}$

## Footnotes and references

1. The Lorentz force law was actually derived by Maxwell under the name of "Equation for Electromotive Force" and was one of an earlier set of eight Maxwell's equations. See the "History" section.
2. David J Griffiths (1999). Introduction to electrodynamics (Third Edition ed.). Prentice Hall. p. pp. 559-562. ISBN 013805326X.
3. Ironically it is an equation which Maxwell himself was absolutely responsible for even though it doesn't count as a "Maxwell's equation". This extra equation appeared in an original list of eight Maxwell's equations in his 1865 paper entitled A Dynamical Theory of the Electromagnetic Field. Maxwell derived it from Faraday's law when Lorentz was still a young boy and he termed it the equation of electromotive force.
4. Oliver Heaviside ((2001) Facsimile of 1893 Edition). Electromagnetic theory. Adamant Media Corporation. p. Vol. 1. ISBN 1402172982. Check date values in: |date= (help)
5. Oliver Heaviside ((2007) Facsimile of 1912 Edition). Electromagnetic theory. Cosimo Classics. p. Vol. 3. ISBN 1602062625. Check date values in: |date= (help)
7. See NIST Special Publication 330, Appendix 2, p. 45 : "Current practice is to use c0 to denote the speed of light in vacuum (ISO 31)."
8. These complications show there is merit in separating the Lorentz force from the main four Maxwell equations. The four Maxwell's equations express the fields' dependence upon current and charge, setting apart the calculation of these currents and charges. As noted in this subsection, these calculations may well involve the Lorentz force only implicitly. Separating these complicated considerations from the Maxwell's equations provides a useful framework.
9. D. E. Aspnes, "Local-field effects and effective-medium theory: A microscopic perspective," Am. J. Phys. 50, p. 704-709 (1982).
10. Habib Ammari & Hyeonbae Kang (2006). Inverse problems, multi-scale analysis and effective medium theory : workshop in Seoul, Inverse problems, multi-scale analysis, and homogenization, June 22-24, 2005, Seoul National University, Seoul, Korea. Providence RI: American Mathematical Society. ISBN 0821839683.
11. O. C. Zienkiewicz, Robert Leroy Taylor, J. Z. Zhu, Perumal Nithiarasu (2006). The Finite Element Method (Sixth Edition ed.). Oxford UK: Butterworth-Heinemann. p. pp. 550 ff. ISBN 0750663219.
12. See, e.g.: N. Bakhvalov and G. Panasenko, Homogenization: Averaging Processes in Periodic Media (Kluwer: Dordrecht, 1989); V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer: Berlin, 1994).
13. Vitaliy Lomakin, Steinberg BZ, Heyman E, & Felsen LB (2003). "Multiresolution Homogenization of Field and Network Formulations for Multiscale Laminate Dielectric Slabs" (PDF). IEEE Transactions on Antennas and Propagation. 51 (10): 2761 ff. doi:10.1109/TAP.2003.816356.
14. AC Gilbert (Ronald R Coifman, Editor) (2000). Topics in Analysis and Its Applications: Selected Theses. Singapore: World Scientific Publishing Company. p. 155. ISBN 9810240945.
15. Edward D. Palik & Ghosh G (1998). Handbook of Optical Constants of Solids. London UK: Academic Press. ISBN 0125444222.
16. F Capasso, JN Munday, D. Iannuzzi & HB Chen Casimir forces and quantum electrodynamical torques: physics and nanomechanics
17. Peter Monk (2003). Finite Element Methods for Maxwell's Equations. Oxford UK: Oxford University Press. p. pp. 1 ff. ISBN 0198508883.
18. Thomas B. A. Senior & John Leonidas Volakis (1995). Approximate Boundary Conditions in Electromagnetics. London UK: Institution of Electrical Engineers. p. pp. 261 ff. ISBN 0852968493.
19. T Hagstrom (Björn Engquist & Gregory A. Kriegsmann, Eds.) (1997). Computational Wave Propagation. Berlin: Springer. p. pp. 1 ff. ISBN 0387948740.
20. Henning F. Harmuth & Malek G. M. Hussain (1994). Propagation of Electromagnetic Signals. Singapore: World Scientific. p. p. 17. ISBN 9810216890.
21. S. F. Mahmoud (1991). Electromagnetic Waveguides: Theory and Applications applications. London UK: Institution of Electrical Engineers. p. Chapter 2. ISBN 0863412327.
22. Jean-Michel Lourtioz (2005). Photonic Crystals: Towards Nanoscale Photonic Devices. Berlin: Springer. p. pp. 84. ISBN 354024431X.
23. S. G. Johnson, Notes on Perfectly Matched Layers, online MIT course notes (Aug. 2007).
24. Taflove A & Hagness S C (2005). Computational Electrodynamics: The Finite-difference Time-domain Method. Boston MA: Artech House. p. Chapters 6 & 7. ISBN 1580538320.
25. David M Cook (2003). The Theory of the Electromagnetic Field. Mineola NY: Courier Dover Publications. p. pp. 335 ff. ISBN 0486425673.
26. Korada Umashankar (1989). Introduction to Engineering Electromagnetic Fields. Singapore: World Scientific. p. §10.7; pp. 359ff. ISBN 9971509210.
27. Joseph V. Stewart (2001). Intermediate Electromagnetic Theory. Singapore: World Scientific. p. Chapter III, pp. 111 ff Chapter V, Chapter VI. ISBN 9810244703.
28. Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. p. pp. 333ff and Chapter 3: pp. 89ff. ISBN 0-7637-3827-1.
29. John Leonidas Volakis, Arindam Chatterjee & Leo C. Kempel (1989). Finite element method for electromagnetics : antennas, microwave circuits, and scattering applications. New York: Wiley IEEE. p. pp. 79 ff. ISBN 0780334256.
30. Bernard Friedman (1990). Principles and Techniques of Applied Mathematics. Mineola NY: Dover Publications. ISBN 0486664449.
31. The term Maxwell-Faraday equation frequently is replaced by Faraday's law of induction or even Faraday's law. These last two terms have multiple meanings, so Maxwell-Faraday equation is used here to avoid confusion.
32. Details can be found in U. Krey, A. Owen, Basic Theoretical Physics - A Concise Overview, Springer, Berlin and elsewhere, 2007, ISBN 978-3-540-36804-5
33. Jackson, Section 6.2

### Journal articles

The developments before relativity

• Joseph Larmor (1897) "On a dynamical theory of the electric and luminiferous medium", Phil. Trans. Roy. Soc. 190, 205-300 (third and last in a series of papers with the same name).
• Hendrik Lorentz (1899) "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam, I, 427-43.
• Hendrik Lorentz (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam, IV, 669-78.
• Henri Poincaré (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaies, V, 253-78.
• Henri Poincaré (1901) Science and Hypothesis
• Henri Poincaré (1905) "Sur la dynamique de l'electron", Comptes Rendues, 140, 1504-8.

see

### University level textbooks

• Tipler, Paul; Mosca, Gene (2007), Physics for Scientists and Engineers: Volume 2 (6th ed.), W. H. Freeman, ISBN 978-1429201339
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (2008), Foundations of Electromagnetic Theory (4th ed.), Addison Wesley, ISBN 978-0321581747
• Edward Mills Purcell (1985). Electricity and Magnetism. McGraw-Hill. ISBN 0-07-004908-4.
• Schwarz, Melvin (1987). Principles of Electrodynamics. Dover Publications. ISBN 0-486-65493-1.
• Stevens, Charles F., 1995. The Six Core Theories of Modern Physics. MIT Press. ISBN 0-262-69188-4.
• Krey, U., Owen, A. (2007), Basic Theoretical Physics - A Concise Overview, esp. part II, Springer, ISBN 978-3-540-36804-5
• Ulaby, Fawwaz T. (2007). Fundamentals of Applied Electromagnetics (5th ed.). Pearson Education, Inc. ISBN 0-13-241326-4.
• Sadiku, Matthew N. O. (2006). Elements of Electromagnetics (4th ed.). Oxford University Press. ISBN 0-19-5300483.
• Hoffman, Banesh, 1983. Relativity and Its Roots. W. H. Freeman.

• Panofsky, Wolfgang K. H.; Phillips, Melba (2005/1962), Classical Electricity and Magnetism (2nd ed.), Dover, ISBN 978-0486439242 Check date values in: |date= (help)